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# 3.4 derivative and graphs

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### 3.4 derivative and graphs

1. 1. Derivatives and Graphs
2. 2. Derivatives and Graphs Interval Notation We use “(” and “)” to indicate the corresponding point is excluded and we use “[” and “]” to for the inclusion of the point. a < x < b (a, b) a b a closed interval: a ≤ x ≤ b a b a ≤ x < b a [a, b) b [a, b] a < x ≤ b a b (a, b] For the unbounded intervals, use “(” or “) for ±∞, for example, “(–∞, a] or (a, ∞) are the following intervals. x < a (–∞, a] a –∞ a < x a ∞ an open interval: half–open intervals:
3. 3. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x).
4. 4. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”.
5. 5. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I y = f(x) ) I (
6. 6. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. y = f(x) (x, f(x)) I ( )
7. 7. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. y = f(x) (x, f(x)) I This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) ( )
8. 8. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
9. 9. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
10. 10. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
11. 11. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
12. 12. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) goes to the same limit as those connecting to the left of x as h goes to 0. y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
13. 13. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) goes to the same limit as those connecting to the left of x as h goes to 0. y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
14. 14. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) goes to the same limit as those connecting to the left of x as h goes to 0. y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
15. 15. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) goes to the same limit as those connecting to the left of x as h goes to 0. y = f(x) (x, f(x)) (x+h, f(x+h)) I ( )
16. 16. Derivatives and Graphs In next few sections, we use the derivative y' to obtain information about the graph of function y = f(x). To see what information the derivative of a function gives us, we need to take a closer look at the notion of “the tangent line”. Let y = f(x) be continuous in an open interval I and f '(x) = lim Δy/Δx exists at some generic point x in I. This means the slopes of the cords on the right side of the base point (x, f(x)) connecting to (x+h, f(x+h)) goes to the same limit as those connecting to the left of x as h goes to 0. y = f(x) (x, f(x)) (x+h, f(x+h)) Cords from the right and left merge into one “tangent line” I ( )
17. 17. Derivatives and Graphs Hence the existence of the f '(x), i.e. having a slope, guarantees a seamless joint at (x, f(x)) from the two sides which corresponds to our notion of “smooth” at P. Therefore f '(x) exists ↔ the graph of y = f(x) is smooth at the point (x, f(x)). y = f(x) (x, f(x)) and f '(x) exists For example, if the left side cords and the right side cords converge to different lines, then there is a corner at P. hence the graph is not smooth at P. P(x, f(x)) However, if lim Δy/Δx does not exist then there are multiple possibilities. y = f(x) f '(x) fails to exist at P
18. 18. Derivatives and Graphs Let’s examine closer the geometry of the graph given that f '(x) exists. Given that f '(x) = 0, i.e. the tangent line is flat at (x, f(x)), there are four possible shapes of y = f(x). (x, f(x)) the graph y = f(x) crosses the tangent line Given that f '(x) > 0, i.e. the slope is positive at (x, f(x)), there are four possible shapes of y = f(x). the graph y = f(x) stays on the same side of the tangent line the graph y = f(x) crosses the tangent line the graph y = f(x) stays on the same side of the tangent line (x, f(x)) (x, f(x)) (x, f(x)) (x, f(x)) (x, f(x)) (x, f(x)) (x, f(x)) Draw the four possible graphs if f '(x) < 0.
19. 19. Derivatives and Graphs Given the graph below, we can easily identify the points whose tangents are horizontal lines. y = f(x) a f
20. 20. Derivatives and Graphs Given the graph below, we can easily identify the points whose tangents are horizontal lines. y = f(x) B C D E a f
21. 21. Derivatives and Graphs Given the graph below, we can easily identify the points whose tangents are horizontal lines. B C D E y = f(x) a f A point P(x, y) is a critical point if f '(x) = 0, or that f '(x) is undefined (more on this later).
22. 22. Derivatives and Graphs Given the graph below, we can easily identify the points whose tangents are horizontal lines. B C D E y = f(x) a f A point P(x, y) is a critical point if f '(x) = 0, or that f '(x) is undefined (more on this later). The critical points where f '(x) = 0 are the “flat–points”.
23. 23. Derivatives and Graphs Given the graph below, we can easily identify the points whose tangents are horizontal lines. b B C D E y = f(x) a c d e f A point P(x, y) is a critical point if f '(x) = 0, or that f '(x) is undefined (more on this later). The critical points where f '(x) = 0 are the “flat–points”. So B, C, D, and E are flat (critical) points and that f '(b) = f '(c) = .. = 0.
24. 24. Derivatives and Graphs y = f(x) the absolute maximum in the interval (a, f) A b B C D E F I a c d e f The point C above is the absolute maximum in the interval I = (a, f).
25. 25. Derivatives and Graphs y = f(x) the absolute maximum in the interval (a, f) A b B C D E F I a c d e f The point C above is the absolute maximum in the interval I = (a, f). Given a function y = f(x) with domain I and that u ϵ I, we say that (u, f(u)) is an absolute maximum in I if f(u) ≥ f(x) for all x’s in I.
26. 26. Derivatives and Graphs y = f(x) the absolute maximum in the interval (a, f) A b B C D E F I a c d e f The point C above is the absolute maximum in the interval I = (a, f). Given a function y = f(x) with domain I and that u ϵ I, we say that (u, f(u)) is an absolute maximum in I if f(u) ≥ f(x) for all x’s in I. An absolute maximum is not lower than any other point on the graph.
27. 27. Derivatives and Graphs y = f(x) the absolute maximum in the interval (a, f) A b B C D E F I a c d e f The point C above is the absolute maximum in the interval I = (a, f). Given a function y = f(x) with domain I and that u ϵ I, we say that (u, f(u)) is an absolute maximum in I if f(u) ≥ f(x) for all x’s in I. An absolute maximum is not lower than any other point on the graph. y = sin(x) has infinitely many absolute maxima. y = sin(x)
28. 28. b B C D E A a c d e f F Derivatives and Graphs However if the graph continues on and C is not the overall highest point, than we say C is a “local” maximum as shown here. C a f the absolute maximum in the interval (a, f) y = f(x)
29. 29. b B C D E A a c d e f F Derivatives and Graphs However if the graph continues on and C is not the overall highest point, than we say C is a “local” maximum as shown here. C a f g the absolute maximum in the interval (a, f) the graph is higher here y = f(x)
30. 30. b B C D E A a c d e f F Derivatives and Graphs However if the graph continues on and C is not the overall highest point, than we say C is a “local” maximum as shown here. C a local maximum in the interval (a, g) a f g the absolute maximum in the interval (a, f) the graph is higher here y = f(x)
31. 31. b B C D E A a c d e f F Derivatives and Graphs However if the graph continues on and C is not the overall highest point, than we say C is a “local” maximum as shown here. In general, we say that (u, f(u)) is a local maximum if C f(u) ≥ f(x) for all x’s in some open neighborhood N in the a local maximum domain as shown here. in the interval (a, g) N g a f the absolute maximum in the interval (a, f) the graph is higher here y = f(x)
32. 32. b B C D the absolute minimum at x = e E y = f(x) A a c d e f F Derivatives and Graphs The point E above is the absolute minimum in (a, f).
33. 33. b B C D the absolute minimum at x = e E y = f(x) A a c d e f F Derivatives and Graphs The point E above is the absolute minimum in (a, f). Similarly we say that (v, f(v)) is an absolute minimum if f(x) ≥ f(v) for all x’s in the domain I.
34. 34. b B C D the absolute minimum at x = e E y = f(x) A a c d e f F Derivatives and Graphs The point E above is the absolute minimum in (a, f). Similarly we say that (v, f(v)) is an absolute minimum if f(x) ≥ f(v) for all x’s in the domain I. An absolute minimum is not lower than any other point on the graph.
35. 35. b B C D the absolute minimum at x = e E y = f(x) A a c d e f F Derivatives and Graphs The point E above is the absolute minimum in (a, f). Similarly we say that (v, f(v)) is an absolute minimum if f(x) ≥ f(v) for all x’s in the domain I. y = tan(x) An absolute minimum is not lower than any other point on the graph. Note that y = tan(x) does not have any extremum in the interval (–π/2, π/2). –π/2 π/2
36. 36. Derivatives and Graphs b B A a E e f F C c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I.
37. 37. Derivatives and Graphs b B increasing A a E e f F C c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I. The curve above is increasing in (a, b),
38. 38. Derivatives and Graphs increasing b B increasing A a increasing E e f F C c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I. The curve above is increasing in (a, b), (b, c) and (e, f).
39. 39. Derivatives and Graphs increasing b B increasing A a increasing E e f F C u < v c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I. The curve above is increasing in (a, b), (b, c) and (e, f). If f(x) is increasing in an interval I, and u, v are any two points in I with u < v,
40. 40. Derivatives and Graphs increasing b B increasing A a increasing E e f F C f(u)< f(v) u < v c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I. The curve above is increasing in (a, b), (b, c) and (e, f). If f(x) is increasing in an interval I, and u, v are any two points in I with u < v, then f(u) < f(v).
41. 41. Derivatives and Graphs increasing b B increasing A a increasing E e f F C f(u)< f(v) u < v c We say that the function is increasing at x if f '(x) > 0 and we say that f (x) is increasing in an open interval I if f '(x) > 0 for all x’s in I. The curve above is increasing in (a, b), (b, c) and (e, f). If f(x) is increasing in an interval I, and u, v are any two points in I with u < v, then f(u) < f(v). We say that f(x) is non–decreasing if f(u) ≤ f(v) for any u < v.
42. 42. Derivatives and Graphs C c D E d e We say that the function is decreasing at x if f '(x) < 0 and we say that f (x) is decreasing in an open interval I if f '(x) < 0 for all x’s in I.
43. 43. Derivatives and Graphs decreasing C decreasing c D E d e We say that the function is decreasing at x if f '(x) < 0 and we say that f (x) is decreasing in an open interval I if f '(x) < 0 for all x’s in I. The curve above is decreasing in (c, d), and (d, e).
44. 44. Derivatives and Graphs decreasing C decreasing c D E f(u) > f(v) d u < v e We say that the function is decreasing at x if f '(x) < 0 and we say that f (x) is decreasing in an open interval I if f '(x) < 0 for all x’s in I. The curve above is decreasing in (c, d), and (d, e). If f(x) is decreasing in an interval I, and u, v are any two points in I with u < v, then f(u) > f(v).
45. 45. Derivatives and Graphs decreasing C decreasing c D E f(u) > f(v) d u < v e We say that the function is decreasing at x if f '(x) < 0 and we say that f (x) is decreasing in an open interval I if f '(x) < 0 for all x’s in I. The curve above is decreasing in (c, d), and (d, e). If f(x) is decreasing in an interval I, and u, v are any two points in I with u < v, then f(u) > f(v). We say that f(x) is non–increasing if f(u) ≥ f(v) for any u < v.
46. 46. Derivatives and Graphs Summary of the graphs given the sign of f '(x). f '(x) = slope = 0 max, min, flat–landing point (x, f(x)) Draw them. (x, f(x)) (x, f(x)) (x, f(x)) f '(x) = slope > 0 increasing, going uphill (x, f(x)) (x, f(x)) (x, f(x)) (x, f(x)) f '(x) = slope < 0 decreasing, going downhill Let’s apply the above observations to the monomial functions y = xN where N = 2,3,4..
47. 47. Derivatives and Graphs The graphs y = xeven The derivative of y = xeven is y' = #xodd. So if x < 0, y' < 0, the function is decreasing, and if x > 0, y > 0 and y is increasing. y = x6 y = x4 y = x2 (-1, 1) (1, 1) y' < 0 y' > 0 (0,0) (0, 0) is the absolute min.
48. 48. Derivatives and Graphs The graphs y = xodd The graphs y = xeven The derivative of y = xeven is y' = #xodd. So if x < 0, y' < 0, the function is decreasing, and if x > 0, y > 0 and y is increasing. The derivative of y = xodd is y' = #xeven. For x ≠ 0, y' > 0, so the function is increasing where x ≠ 0. y = x5 y = x3 y = x7 (1, 1) (-1, -1) y = x6 y = x4 y = x2 (-1, 1) (1, 1) (0,0) y' < 0 y' > 0 (0,0) y' > 0 except at y'(0) = 0 (0, 0) is the absolute min. (0, 0) is a flat–landing.
49. 49. Derivatives and Graphs Here are the general steps for graphing.
50. 50. Derivatives and Graphs Here are the general steps for graphing. Steps 1 and 2 do not require calculus. 1. Determine the domain of f(x) and the behavior of y as x approaches the boundary of the domain. 2. Use the roots and asymptotes to make the sign–chart and determine the general shape of the graph.
51. 51. Derivatives and Graphs Here are the general steps for graphing. Steps 1 and 2 do not require calculus. 1. Determine the domain of f(x) and the behavior of y as x approaches the boundary of the domain. 2. Use the roots and asymptotes to make the sign–chart and determine the general shape of the graph. Step 3 and 4 uses the 1st derivative of f(x). 3. Find the derivative f '(x), use the roots of f '(x) = 0 to find the extrema and flat–points. 4. Make the sign–chart of f '(x) to determine the terrain of y = f (x), i.e. the graph is going uphill where f '(x) > 0 and downhill where f '(x) < 0.
52. 52. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials.
53. 53. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials. Example A. Graph y = 9x – x3. Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing?
54. 54. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials. Example A. Graph y = 9x – x3. Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing? We start by factoring to locate the roots. 9x – x3 = x(3 – x)(3 + x), hence x = 0, 3, –3 are the roots and each is of order 1.
55. 55. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials. Example A. Graph y = 9x – x3. Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing? We start by factoring to locate the roots. 9x – x3 = x(3 – x)(3 + x), hence x = 0, 3, –3 are the roots and each is of order 1. The sign–chart and the graph of f(x) is shown here.
56. 56. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials. Example A. Graph y = 9x – x3. Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing? We start by factoring to locate the roots. 9x – x3 = x(3 – x)(3 + x), hence x = 0, 3, –3 are the roots and each is of order 1. The sign–chart and the graph of f(x) is shown here. x y + – + – –3 0 3 y = 9x – x3
57. 57. Derivatives and Graphs The domain of polynomials is the set of all real numbers. Polynomial graphs are smooth everywhere because derivatives of polynomials are well defined polynomials. Example A. Graph y = 9x – x3. Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing? We start by factoring to locate the roots. x y + – + – –3 0 3 y = 9x – x3 9x – x3 = x(3 – x)(3 + x), hence x = 0, 3, –3 are the roots and each is of order 1. The sign–chart and the graph of f(x) is shown here.
58. 58. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2. x y y = 9x – x3 + – + – –3 0 3
59. 59. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2. Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 x y y = 9x – x3 + – + – –3 0 3
60. 60. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2 Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 x y + – + – –3 0 3 the corresponding points on the graphs are (√3, 9√3 – (√3)3) = (√3, 6√3) and (–√3, –6√3) which are the unique local max. and the local min. respectively. y = 9x – x3
61. 61. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2 Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 x y + – + – –3 0 3 the corresponding points on the graphs are (√3, 9√3 – (√3)3) = (√3, 6√3) and (–√3, –6√3) which are the unique local max. and the local min. respectively. y = 9x – x3 (√3, 6√3) (–√3, –6√3) Label these points on the graph.
62. 62. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2 Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 is shown below the graph. x y + – + – –3 0 3 the corresponding points on the graphs are (√3, 9√3 – (√3)3) = (√3, 6√3) and (–√3, –6√3) which are the unique local max. and the local min. respectively. y = 9x – x3 (√3, 6√3) (–√3, –6√3) Label these points on the graph. The sign–chart of the y' = 9 – 3x2 = 3(3 – x2) = 3(x – √3)(x + √3)
63. 63. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2 Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 is shown below the graph. x y + – + – –3 0 3 the corresponding points on the graphs are (√3, 9√3 – (√3)3) = (√3, 6√3) and (–√3, –6√3) which are the unique local max. and the local min. respectively. y = 9x – x3 (√3, 6√3) (–√3, –6√3) –√3 √3 Label these points on the graph. The sign–chart of the y' = 9 – 3x2 = 3(3 – x2) = 3(x – √3)(x + √3) y' = -3(x–√3)(x+√3)
64. 64. Derivatives and Graphs The derivative is y' = (9x – x3)' = 9 – 3x2 Set y' = 9 – 3x2 = 3(3 – x2) = 0  x = ±√3 is shown below the graph. x y + – + – –3 0 3 the corresponding points on the graphs are (√3, 9√3 – (√3)3) = (√3, 6√3) and (–√3, –6√3) which are the unique local max. and the local min. respectively. y = 9x – x3 (√3, 6√3) (–√3, –6√3) –√3 √3 Label these points on the graph. The sign–chart of the y' = 9 – 3x2 = 3(3 – x2) = 3(x – √3)(x + √3) y' = -3(x–√3)(x+√3) y' is + uphill y' is – downhill y' is – downhill
65. 65. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill Max up down
66. 66. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Max up down down up Min
67. 67. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Hence the signs of the y' must change at these critical points. Max up down down up Min
68. 68. Derivatives and Graphs Max up down At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Hence the signs of the y' must change at these critical points. Both f(x) = x3(3 – x)(3 + x) = x3(9 – x2) = 9x3 – x5 and g(x) = x(9 – x2) have the same sign–chart so both graphs are similar. down up Min
69. 69. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Hence the signs of the y' must change at these critical points. up down down up Both f(x) = x3(3 – x)(3 + x) = x3(9 – x2) = 9x3 – x5 and g(x) = x(9 – x2) have the same sign–chart so both graphs are similar. However, for f(x) it’s derivative f '(x) = [9x3 – x5 ]' = 27x2 – 5x4 = x2(27 – 5x2). Max Min
70. 70. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Hence the signs of the y' must change at these critical points. up down down up Both f(x) = x3(3 – x)(3 + x) = x3(9 – x2) = 9x3 – x5 and g(x) = x(9 – x2) have the same sign–chart so both graphs are similar. However, for f(x) it’s derivative f '(x) = [9x3 – x5 ]' = 27x2 – 5x4 = x2(27 – 5x2). signs around x = 0 of f '(x) = x2(27 – 5x2) + uphill + uphill f '(0)=0 with order 2 Max Min
71. 71. Derivatives and Graphs At a maximum point the graph must be changing from going uphill to going downhill and at the minimum point it must be changing from going downhill to uphill. Hence the signs of the y' must change at these critical points. Max up down down up Min Both f(x) = x3(3 – x)(3 + x) = x3(9 – x2) = 9x3 – x5 and g(x) = x(9 – x2) have the same sign–chart so both graphs are similar. However, for f(x) it’s derivative f '(x) = [9x3 – x5 ]' = 27x2 – 5x4 = x2(27 – 5x2). + uphill + uphill f '(0)=0 with order 2 signs around x = 0 of f '(x) = x2(27 – 5x2) The sign of f '(x) is positive on both sides of x = 0, so (0, 0) is not a max nor min, it’s a flat landing point.
72. 72. Derivatives and Graphs We put the two graphs side by side for comparison. y = g(x) = 9x – x3 y = f(x) = x2g(x) = 9x3 – x5 x y + – + – –3 0 3 x + – + – –3 0 3 f(0) = 0 of order 3, and f'(0) = 0 of order 2, flat point f(0) = 0 of order 1, and f'(0) = 9 > 0, uphill y
73. 73. Derivatives and Graphs We put the two graphs side by side for comparison. y = g(x) = 9x – x3 y = f(x) = x2g(x) = 9x3 – x5 x y + – + – –3 0 3 x + – + – –3 0 3 f(0) = 0 of order 3, and f'(0) = 0 of order 2, flat point f(0) = 0 of order 1, and f'(0) = 9 > 0, uphill y So if f '(x) = 0 of even order, then it’s a flat landing point, not a max nor min.
74. 74. Derivatives and Graphs We put the two graphs side by side for comparison. y = g(x) = 9x – x3 y = f(x) = x2g(x) = 9x3 – x5 x y + – + – –3 0 3 x + – + – –3 0 3 f(0) = 0 of order 3, and f'(0) = 0 of order 2, flat point f(0) = 0 of order 1, and f'(0) = 9 > 0, uphill y So if f '(x) = 0 of even order, then it’s a flat landing point, not a max nor min. Example B. Graph y = x + sin(x). Find the locations of the extrema, the flat points and classify them. For what x values is y increasing and for what x values is y decreasing?
75. 75. Derivatives and Graphs We observe that f(x) = x + sin(x) = 0 if x = 0.
76. 76. Derivatives and Graphs We observe that f(x) = x + sin(x) = 0 if x = 0. Furthermore, f(x) > 0 if x > 0, f(x) < 0 if x < 0 (why?), and as x →±∞, f(x) →±∞
77. 77. Derivatives and Graphs We observe that f(x) = x + sin(x) = 0 if x = 0. Furthermore, f(x) > 0 if x > 0, f(x) < 0 if x < 0 (why?), and as x →±∞, f(x) →±∞ The derivative f '(x) = 1 + cos(x) = 0 so the critical points f '(x) = 0 are at x = ±π, ±3 π, .. where cos(x) = –1.
78. 78. Derivatives and Graphs We observe that f(x) = x + sin(x) = 0 if x = 0. Furthermore, f(x) > 0 if x > 0, f(x) < 0 if x < 0 (why?), and as x →±∞, f(x) →±∞ The derivative f '(x) = 1 + cos(x) = 0 so the critical points f '(x) = 0 are at x = ±π, ±3 π, .. where cos(x) = –1. x y = f(x) π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π some points where f '(x) = 0 In particular f(π) = π, f(3π) = 3π, etc.. so the flat points of y = f(x) are at (x, x) with x = ± π, ±3 π, ±5 π..
79. 79. Derivatives and Graphs We observe that f(x) = x + sin(x) = 0 if x = 0. Furthermore, f(x) > 0 if x > 0, f(x) < 0 if x < 0 (why?), and as x →±∞, f(x) →±∞ The derivative f '(x) = 1 + cos(x) = 0 so the critical points f '(x) = 0 are at x = ±π, ±3 π, .. where cos(x) = –1. x y = f(x) π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π some points where f '(x) = 0 In particular f(π) = π, f(3π) = 3π, etc.. so the flat points of y = f(x) are at (x, x) with x = ± π, ±3 π, ±5 π.. Furthermore the derivative f '(x) = 1 + cos(x) > 0 so f(x) = x + sin(x) is increasing everywhere else. Let’s putting it all together.
80. 80. Derivatives and Graphs Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π points where f '(x) = 0
81. 81. Derivatives and Graphs x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0) points where f '(x) = 0
82. 82. Derivatives and Graphs x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . The graph is a continuous smooth curve that is increasing between all the horizontal locations. (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0) points where f '(x) = 0
83. 83. Derivatives and Graphs x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . The graph is a continuous smooth curve that is increasing between all the horizontal locations. The only possibility for a smooth increasing curve between two flat–spots is shown here. (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0) points where f '(x) = 0
84. 84. Derivatives and Graphs Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . The graph is a continuous smooth curve that is increasing between all the horizontal locations. The only possibility for a smooth increasing curve between two flat–spots is shown here. x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0) points where f '(x) = 0
85. 85. Derivatives and Graphs The graph is a continuous smooth curve that is increasing between all the horizontal locations. The only possibility for a smooth increasing curve between two flat–spots is shown here. x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . points where f '(x) = 0 (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0)
86. 86. Derivatives and Graphs The graph is a continuous smooth curve that is increasing between all the horizontal locations. The only possibility for a smooth increasing curve between two flat–spots is shown here. x y π π 3π 3π 5π 5π –π –π –3π –3π –5π –5π Here is a table of some of the points where y’ = 0, i.e. their tangents are horizontal . points where f '(x) = 0 (π, π) (3π, 3π) (–π, –π) (–3π, –3π) (0, 0) Hence the graph of y = x + sin(x) is:
87. 87. Derivatives and Graphs The critical points where f '(x) fails to exist are “corners”, or at points where the tangent line is vertical. Questions a. Find f '(x) as x → 0+, and as x → 0– for each of the following function. b. How does their graphs at x = 0 reflect the answers from a? f (x) = | x | f (x) = x2/3 f (x) = x1/3